Problem: Two friends are playing tic-tac-toe. If Amy wins $\frac{3}{8}$ of the time, Lily wins $\frac{3}{10}$ of the time, and they tie the rest of the time, then what fraction of the time do they tie?
Answer: We first find the fraction of the time that either Amy or Lily wins by adding $\frac{3}{8} + \frac{3}{10}$. Since the least common multiple of $8$ and $10$ is $40$, we write $\frac{3}{8} \cdot \frac{5}{5} = \frac{15}{40}$ and $\frac{3}{10} \cdot \frac{4}{4} = \frac{12}{40}$, so our sum is: $$\frac{3}{8} + \frac{3}{10} = \frac{15}{40} + \frac{12}{40} = \frac{15+12}{40} = \frac{27}{40}.$$Since they tie the rest of the time, to find the fraction of the time that they tie, we need to subtract this fraction from $1$. We know that $1 = \frac{40}{40}$, so we have that $$1 - \frac{27}{40} = \frac{40}{40} - \frac{27}{40} = \frac{40-27}{40} = \frac{13}{40}.$$Therefore, Amy and Lily must tie $\boxed{\frac{13}{40}}$ of the time.